## Abstract

Atmospheric turbulence corrupts astronomical images formed by ground-based telescopes. Adaptive optics systems allow the effects of turbulence-induced aberrations to be reduced for a narrow field of view corresponding approximately to the isoplanatic angle
${\theta}_{0}$. For field angles larger than
${\theta}_{0}$, the point spread function (PSF) gradually degrades as the field angle increases.
We present a technique to estimate the PSF of an adaptive optics telescope as function of the field angle, and use this information in a space-varying image reconstruction technique. Simulated anisoplanatic intensity images of a star field are reconstructed by means of a block-processing method using the predicted local PSF. Two methods for image recovery are used:
matrix inversion with Tikhonov regularization, and the Lucy–Richardson algorithm. Image reconstruction results obtained using the space-varying predicted PSF are compared to space invariant deconvolution results obtained using the on-axis PSF. The anisoplanatic reconstruction technique using the predicted PSF provides a significant improvement of the mean squared error between the reconstructed image and the object compared to the deconvolution performed using the on-axis PSF.

© 2007 Optical Society of America

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### Equations (14)

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(1)
$${\theta}_{0}=\frac{58.1\times {10}^{-3}{\lambda}^{6/5}}{{\left[{\left(\mathrm{sec}\text{\hspace{0.17em}}{\theta}_{z}\right)}^{8/3}{\displaystyle {\int}_{0}^{L}\mathrm{d}z{{C}_{n}}^{2}\left(z\right){z}^{5/3}}\right]}^{3/5}}\text{,}$$
(2)
$${r}_{0}=0.185{\left[\frac{4{\pi}^{2}}{\mathrm{sec}\text{\hspace{0.17em}}{\theta}_{z}{k}^{2}{\displaystyle {\int}_{{z}_{1}}^{{z}_{2}}\mathrm{d}z{{C}_{n}}^{2}\left(z\right)}}\right]}^{3/5}\text{,}$$
(3)
$${{c}_{\perp \text{,}\theta}}^{\mathrm{mod}}\left({u}_{\perp}\right)={\displaystyle \sum _{i=0}^{N}{a}_{i}{T}_{i}\left({u}_{\perp}\right)\text{\hspace{0.17em} for \hspace{0.17em}}{u}_{\perp}\ge 0}\text{,}$$
(4)
$${{c}_{\theta}}^{\mathrm{mod}}\left(u\right)={\displaystyle \sum _{i=0}^{N}{b}_{i}{T}_{i}\left(u\right)\text{\hspace{0.17em} for \hspace{0.17em}}u\ge 0}\text{,}$$
(5)
$${{h}_{\theta}}^{\mathrm{mod}}\left(\rho ,\beta \right)=\left[1-\gamma \left(\beta \right)\right]{{c}_{\theta}}^{\mathrm{mod}}\left(\rho \right)+\gamma \left(\beta \right){{c}_{\perp \text{,}\theta}}^{\mathrm{mod}}\left(\rho \right)\text{,}$$
(6)
$${\u03f5}_{\text{fit}}\left(\theta \right)=\frac{{\displaystyle {\sum}_{u}{\displaystyle {\sum}_{{u}_{\perp}}{\Vert {{h}_{\theta}}^{\mathrm{mod}}\left(u,{u}_{\perp}\right)-{h}_{\theta}\left(u,{u}_{\perp}\right)\Vert}^{2}}}}{{\displaystyle {\sum}_{u}{\displaystyle {\sum}_{{u}_{\perp}}{\Vert {h}_{\theta}\left(u,{u}_{\perp}\right)\Vert}^{2}}}}.$$
(7)
$${\overline{K}}_{VM+1}=\frac{{\overline{K}}_{VM}}{2.51}.$$
(8)
$$i={{H}_{\theta}}^{\mathrm{mod}}o+n\text{,}$$
(9)
$${{H}_{\theta}}^{\mathrm{mod}}=\left[\begin{array}{cccc}{H}_{0}& {H}_{{N}_{\text{iso}}-1}& \cdots & {H}_{1}\\ {H}_{1}& {H}_{0}& \cdots & {H}_{2}\\ \vdots & \vdots & \ddots & \vdots \\ {H}_{{N}_{\text{iso}}-1}& {H}_{{N}_{\text{iso}}-2}& \cdots & {H}_{0}\end{array}\right]\text{,}$$
(10)
$${H}_{i}=\left[\begin{array}{cccc}{h}_{i\mathrm{,}0}& {h}_{i\mathrm{,}{N}_{\text{iso}}-1}& \cdots & {h}_{i\mathrm{,}{N}_{\text{iso}}-1}\\ {h}_{i\mathrm{,}1}& {h}_{i\mathrm{,}0}& \cdots & {h}_{i\mathrm{,}{N}_{\text{iso}}-2}\\ \vdots & \vdots & \ddots & \vdots \\ {h}_{i\mathrm{,}{N}_{\text{iso}}-1}& {h}_{i\mathrm{,}{N}_{\text{iso}}-2}& \cdots & {h}_{i\mathrm{,}{N}_{\text{iso}}-0}\end{array}\right]\text{,}$$
(11)
$$\tilde{o}={\left[{\left({{H}_{\theta}}^{\mathrm{mod}}\right)}^{\mathrm{T}}{{H}_{\theta}}^{\mathrm{mod}}+\alpha I\right]}^{-1}{\left({{H}_{\theta}}^{\mathrm{mod}}\right)}^{{\mathrm{T}}_{\mathbf{i}}}\text{,}$$
(12)
$${\tilde{o}}^{\left(k+1\right)}={\tilde{o}}^{\left(k\right)}\times \left[{{h}_{\theta}}^{\mathrm{mod}}\ast \left(\frac{i}{{{h}_{\theta}}^{\mathrm{mod}}\text{\hspace{0.17em}}\ast \text{\hspace{0.17em}}{\tilde{o}}^{\left(k\right)}}\right)\right]\text{,}$$
(13)
$${\u03f5}_{\text{recons}}=\frac{{\displaystyle {\sum}_{x}{\displaystyle {\sum}_{y}{\Vert \tilde{o}\left(x,y\right)-o\left(x,y\right)\Vert}^{2}}}}{{\displaystyle {\sum}_{x}{\displaystyle {\sum}_{y}{\Vert o\left(x,y\right)\Vert}^{2}}}}\text{,}$$
(14)
$$\xi =\frac{{\u03f5}_{\text{on}}-{\u03f5}_{\text{off}}}{{\u03f5}_{\text{on}}}\text{,}$$